Price alternately falling and rising one percent

A stock begins at $100. Its price falls $1\%$, then rises $1\%$, then falls $1\%$, and keeps alternating like this forever. What value does the price approach in the long run?

$0$. Each down-up pair multiplies the price by $0.99\times1.01=0.9999<1$, so after $2n$ moves the price is $100\cdot0.9999^n$, which decays geometrically to zero.